Monday, 16 December 2013

notes vs. tones, digits vs. numbers

There is a little linguistic problem that I've been puzzling over quite a lot recently. It concerns what can probably be best described, in abstract terms, as differentiation between quantity and value in sets.

I first ran into this with number theory, working with the different numerical bases - hexadecimal, binary, decimal - that are relevant in computer science, as I needed to be able to refer to the length of, say, a bit pattern or hexadecimal number separately to the actual values that appeared in it.

In binary this is pretty easy, as there are only 2 possible values. However, if I said 'a byte has 8 digits', and you didn't have the assumed knowledge about binary, it could be misunderstood to mean a byte has 8 values, which would be incorrect, since a byte can only contain some combination of 2 values: 0 and 1. This confusion arises out of the ambiguity of the term digit: does 'digit' refer to the value of an item in the byte, or to the number of items in the byte regardless of their value? In this case, if you know anything about binary, the meaning is obvious, but there are other similar situations where it may not be.

It seems that most often digit is understood to refer to a quantity of items and number is understood to refer to the value of any item. However, all the dictionaries I've checked in seem reluctant to make such a clear distinction, the two words are listed as synonymous, and in reality they are often used interchangeably, making use of either one subject to misinterpretation.

The problem crops up again in music. For example, when we say '4 notes', are we referring to any 4 instances of an item (for example, four B flats), or are we referring specifically to 4 unique values (for example, a set of 10 items in which only four values, say A, F, D and B flat, occur)?
Officially there are separate words to describe quantity and value in music: note is to tone* what digit is to number - the former describes a quantity, the latter a unique value. But again, the two terms get used interchangeably, making it difficult to ensure any description of musical patterns is absolutely unambiguous. What am I missing here?!

In writing this blog post, I made an unsettling discovery: I kept trying to use general words that describe quantity or value, only to discover they could potentially be misinterpreted. Range is one such word. I was initially going to use this term instead of quantity, until I realised it could be misinterpreted in much the same way as digit or number. Are we referring to the number of items in the set, or the number of values occurring in the set? Range is perfectly correct in either context.

Also note how it's almost impossible to discuss these issues without using that pesky term number. In this post I've replaced as many occurrences of the word as possible with quantity, but I've left the ones in the previous paragraph untouched to demonstrate how much we rely on potentially ambiguous language.

One final thought, venturing into even more abstract territory: consider that we say 'number of digits'. This requires some recursive thinking: the digits are a set in which various values, known as numbers, can be stored. Another set, called a number, contains the digits which contain the numbers...see the problem with this terminology?